# Turbulent flow, verifying a general DE solution

#### redbeard

1. The problem statement, all variables and given/known data
Consider a metal sphere of radius r, drag coefficient C, density p_s falling in a fluid p and viscosity n.
Find the acceleration:
I found this to be g - p/p_s * g - (3C/(8*p_s*r))*v^2. Others were in agreeal with this so take it as given.

***Show that your result has the solution v=atanh(bt) and find constants a and b.***

***the part im on
2. Relevant equations

3. The attempt at a solution

So, I let C = g-p/p_s* g since this is a constant term and the same for the ones preceding v^2. I just let P = this stuff.
Thus i have v'=C-P*v^2
Now I have no idea how to solve this DE with the knowledge that I have but the general form of the solution was given.
So I'm guessing there's some sort of other way to verify it, but I'm not sure how.

PS: I'm assuming an initial condition is v(0) = 0 as this was needed in another part.

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#### Mindscrape

That's a good assumption for initial condition.

The best thing to realize is that tanh=sinh/cosh

You know that sinh and cosh are expressed as a combination of exponentials. So find the solution in terms of exponentials, and work the tanh into that solution.

#### Mindscrape

Actually, now that I think about it, the problem is even easier than that. Simply plug tanh in, and show that it works. From there it should just be some algebra to figure out what constant is what.

#### redbeard

Hey Mindscrape,

It turned out I did have the knowledge to solve that DE. I assumed it wasn't separable because when i fed it into Wolfram, Wolfram proceeded to vomit. I guess I shouldn't assume Wolfram is so high and mighty such that if it can't do it then I can't.

Thanks again,
Redbeard

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